In order to fractionate substances by crystallization, a melt is cooled on crystallization surfaces, forming crystals which contain the desired substance. The remaining liquid phase or mother liquor is run off. The crystals are then heated so as to sweat out and run off fractions having lower melting-points. Crystals having a higher melting-point and high purity are left. The purified crystals are then melted, thus obtaining the substance. It has not hitherto been possible to fractionate substances such as paraffins, oils, fats or waxes by this process, since particularly in the "partial melting" or "sweating" phase, there is insufficient adhesion between the vertical crystallization surfaces and the crystallized substances, so that parts of the crystal layer or entire layers become detached and slide off. The sliding-off crystals will then lie on the bottom of the crystallizer and make it difficult or impossible to separate the crystals cleanly from the mother liquor. Even such poorly adhering substances, however, need to be fractionated by modern crystallization techniques. Fractional crystallization has the advantage, inter alia, that it can operate without solvents. It is therefore less expensive and less of a danger to the environment or to human health. It is superior to comparable but outmoded techniques such as removing oil from paraffin by sweating, owing to its shorter working cycles, called "batch times" or "stage times", and since it can be more accurately controlled.